Gender Based Taxation and the Division ofFamily Chores ∗

Alberto Alesinaharvard university

Andrea Ichinouniversity of bologna

Loukas Karabarbounisharvard university

November 9, 2007

Abstract

Gender Based Taxation (GBT) satisfies Ramsey’s optimal criterionby taxing less the more elastic labor supply of (married) women. Thisholds when different elasticities between men and women are taken asexogenous and primitive. But in this paper we also explore differencesin gender elasticities which emerge endogenously in a model in whichspouses bargain over the allocation of home duties. GBT changesspouses’ implicit bargaining power and induces a more balanced al-location of house work and working opportunities between males andfemales. Because of decreasing returns to specialization in home andmarket work, social welfare improves by taxing conditional on gender.When income sharing within the family is substantial, both spousesmay gain from GBT.

JEL-Code: D13, H21, J16, J20.Keywords : Optimal Taxation, Economics of Gender, Family Eco-nomics, Elasticity of Labor Supply.

∗We thank George Akerlof, Stefania Albanesi, George-Marios Angeletos, ClaudiaGoldin, Larry Katz, James Poterba, Emmanuel Saez, Ivan Werning and seminar par-ticipants at the Universities of Berkeley, Bologna, Jerusalem, MIT, Munich, Tsinghua,Instituto Carlo Alberto in Torino, the CEPR-ESSLE and the NBER summer meeting, forhelpful comments and suggestions.

1 Introduction

According to optimal taxation theory a benevolent government should tax

less the goods and services which have a more elastic supply. Women labor

supply is more elastic than men’s. Therefore, tax rates on labor income

should be lower for women than for men.

This argument is well known in the literature, but it is not taken seriously

as a policy proposal. This is surprising since a host of other gender based

policies are routinely discussed, and often implemented, such as gender based

affirmative action, quotas, different retirement policies for men and women,

and also indirect gender based policies like child care subsidies, and maternal

leaves.1 Many of these gender based interventions become even more puzzling

in light of the basic economic principle that society should prefer policies

interfering with “prices” (such as the tax rate) rather than “quantities” (such

as affirmative action or quotas) in the market.2

The optimality of GBT hinges on different elasticities of the labor supply

between men and women. If the labor supply elasticity is taken as a primitive,

exogenous parameter that differentiates genders, then the argument is quite

straightforward. GBT provides substantial welfare and GDP gains because

it minimizes the aggregate social loss from labor market distortions. As we

discuss in more detail later, this argument is robust to perturbations in the

modeling framework (Ramsey or Mirrlees), and goes through in extentions

of the model that consider cross elasticities, heterogenous households and

household production.

However, differences in labor supply functions of men and women, in-

1For instance, gender based affirmative action is common in the US, Spain has recentlyintroduced stringent quotas for female employment in many sectors and public support forchild care is common in many European countries. Sweden has recently reformed paternalleave policies with the goal of inducing males to stay more at home with children andfemales to participate more continuously in the labor market.

2In international trade, for instance, a sort of “folk theorem” states that tariffs areweakly superior to import quotas as a trade policy. Taxing polluting activities is generallyconsidered superior to controlling them with quantitative restrictions.

2

cluding their elasticities may not only depend on innate characteristics or

preferences but emerge endogenously from the internal organization of the

family. In fact, as documented for instance by Goldin (2006), Blau and

Kahn (2007) and Albanesi and Olivetti (2007), both women participation

rate and the elasticity of their labor supply, may evolve over time as a result

of technologically induced or culturally induced changes in the organization

of the family.3 Therefore, we also explore the case in which men and women

are identical in terms of innate abilities, preferences and predispositions, but

men have more explicit bargaining power at home (possibly for historical

reasons). In this case, a gendered allocation of work at home and gender

differences in market participation and labor elasticities derive exclusively

from the intra-family bargaining. If men have a stronger bargaining power,

then they assume fewer unpleasant, tiring home duties. Hence, they par-

ticipate more in the market, exercise more effort, and earn more than their

female spouses. The possibility to avoid home duties allows men to engage

in careers that offer “upside potential” in terms of wages and promotions.

For women, it is the opposite: they basically work only for their wage. As a

result, men are less sensitive to changes in the wage since what matters for

them, relative to women, is also the intrinsic expected pleasure they derive

from careers and market activity. We note that the implied positive corre-

lation between the amount of home duties and the elasticity of labor supply

in our model accords well with recent empirical evidence. Aguiar and Hurst

(2007) and Blau and Kahn (2007) document a decline both in the ratio of

female over male home duty and the ratio of female over male elasticity of

labor supply in the last 50 years.

To the extent that the division of family chores is unbalanced, GBT im-

proves welfare. In addition to satisfying the Ramsey principle of optimal

3Alesina and Giuliano (2007) study the effect of different cultural traits on familyvalues and ties as a determinant of women participation in the labor force. Ichino andMoretti (2006) show instead how more persistent biological gender differences may affectthe absenteeism of men and women and, indirectly, labor market equilibria.

3

taxation, GBT generates a more equitable allocation of house versus market

work. Because of decreasing returns to scale, reallocating “the last hour that

the mother spends with the children to the father” is welfare improving for

the family as a whole, and under certain conditions it can be welfare im-

proving for both spouses (as well as for children). Our numerical simulations

show that given a difference in the labor supply elasticities calibrated on US

estimates (which in the model maps into a corresponding difference in the

bargained allocation of home duties), GBT implies rather different tax rates

for husbands and wives and can substantially improve welfare, as well as

increase GDP and total employment.

We can also interpret our result in terms of a difference between short

versus long run effects of GBT. The case in which labor supply functions and

their different elasticities across gender are exogenous can be interpreted as

the short run, namely an horizon in which the family organization and the

allocation of home duties is not likely to change. In the long run, instead,

the family responds to government policies and evolves to a new equilibrium

with a different organization of allocation of home duties.

We take a different approach from the literature in modeling household

production. The traditional approach builds on the Beckerian theory of the

allocation of time (1965), and assumes that household duty is an input to

the family production function for the production of a household good. In

our model with endogenous gender differences in elasticities we start by a

woman and a man who form a family and receive a collection of shocks that

must be allocated between the two spouses. With this assumption we intend

to capture the fact that there are features of the daily household routine, for

example a sick child or a broken dishwasher, that are easy to conceptualize

as exogenous but negotiable jobs to be done but not as the output of an

intra-household process that transforms time input into a household good.

Obviously the two approaches are not mutually exclusive and a more general

model of household allocation of time and shocks could capture both aspects

4

of family life.

We further illustrate the link between our model and the literature in

Section 2. Section 3 discusses GBT in the short run, that is when gender

differences in labor supply elasticities are held constant. In Section 4 we

endogenize the allocation of household chores and in Section 5 we show how

family bargaining implies an intra-household division of duties, market par-

ticipation and elasticities. In what we call the long run the government sets

taxes anticipating the family’s reaction to fiscal pressure. This is analyzed

in Section 6. Section 7 concludes.

2 Related Literature

The present paper lies at the intersection of three strands of research. The

first is concerned with the structure of the family.4 The traditional “uni-

tary” approach, in the spirit of Samuelson (1956) and Becker (1974), treats

the household as a single decision making unit. Although this approach is

closely linked with the traditional consumer’s theory, it is at odds with the

notion of individualism, and, most importantly for our purposes, lacks the

proper foundations to conduct intrahousehold welfare analysis.5 The “collec-

tive approach” to family modeling, initiated by Chiappori (1988, 1992) and

Apps and Rees (1988), builds instead on the premise that every person has

well defined individual preferences and only postulates that collective deci-

sions lie on the Pareto frontier.6 In fact the (long run) model that we consider

is in the spirit of the collective approach with (asymmetric) Nash-bargained

4See Lundberg and Pollak (1996) and Vermeulen (2002) for excellent surveys.5Two notable empirical failures of the unitary model are the restrictions that arise from

the income pooling hypothesis and the symmetry of the Slutsky matrix. See, for example,Thomas (1990), Browning, Bourguignon, Chiappori and Lechene (1994), Lundberg, Pollakand Wales (1997), and Browning and Chiappori (1998).

6A more specific approach, taken first by Manser and Brown (1980) and McElroy andHorney (1981), “selects” a specific point on the Pareto frontier by assuming that membersof the family Nash-bargain over the allocation of commodities. Lundberg and Pollak(1993), instead, argue that threat points are internal to the marriage and can be seen as(possibly inefficient) non-cooperative equilibria of the marriage game.

5

household allocations. The difference with the above models is that the bar-

gaining is not on the allocation of consumption, income and labor supply per

se, but on the allocation of home duties.

The second relevant strand of literature refers to the taxation of couples.

The “conventional wisdom” says that under specific assumptions, we should

tax at a lower rate goods that are supplied inelastically as suggested by

Ramsey (1927). The application of the Ramsey “inverse elasticity” rule in

a model of labor supply implies that males should be taxed on a higher tax

schedule than females because they have a less elastic labor supply function.

This point was made by Rosen (1977) and Boskin and Sheshinski (1983).7

Since gender is inelastically supplied, this proposition relates also to the

insight that taxes should be conditioned on non-modifiable characteristics as

in Akerlof (1978) and Kremer (2003).8

This conventional wisdom regarding lower taxes for women can be chal-

lenged or reinforced in at least three ways. First, it might be the case that

the female’s tax rate is a better policy instrument when considering across

household redistribution. Boskin and Sheshinski (1983) show that this is not

the case in their numerical calculations. Recently, Apps and Rees (2007)

place the conventional wisdom on a firmer basis and give intuitive and em-

pirically plausible conditions under which it is optimal to tax males at a

higher rate even with heterogeneous households. Second, Piggott and Whal-

ley (1996) raise the issue of intrahousehold distortion of efficiency in models

7The argument was raised using variants of the Diamond and Mirrlees (1971a and1971b) and Atkinson and Stiglitz (1972) frameworks, also adopted in this paper. Usingthe Mirrlees (1971) approach, the elasticity of labor supply reappears in the optimal taxschedule, albeit in a less clear way. For an ambitious paper that takes the latter approachsee Kleven, Kreiner and Saez (2006), or Kremer (2003) within an application to the prob-lem of age based taxation. We also note that in a Mirrleesian application of our model,there is one more factor in favor of GBT: since the female distribution of income has moremass concentrated towards the low income levels, its hazard rate is typically higher andtherefore marginal tax rates for females should be lower.

8See Mankiw and Weinzierl (2007) for a recent application of this idea aimed at dis-cussing the validity of the welfarist approach to optimal taxation.

6

with household production. Since the optimal tax schedule must maintain

productive efficiency (Diamond and Mirrlees 1971a), imposing differential

tax treatment distorts the intrahousehold allocation of resources and raises

a further cost for the society. Although the Piggott and Whalley argument

is intuitive, Apps and Rees (1999b) and Gottfried and Richter (1999) show

that the cost of distorting the intra-household allocation of resources cannot

offset the gains from taxing on an individual basis according to the standard

Ramsey principle. We are interested in exploring the optimality of individ-

ual taxes in a model where within household redistribution is explicitly taken

into account. In that respect our (long run) model is in line and reinforces

the conventional wisdom.9

The third strand of literature attempts to explain gender differences in la-

bor markets. For example, Albanesi and Olivetti (2006) propose that gender

differences can be supported by firms’ expectations that the economy is on a

gendered equilibrium in a model with incentive constraints. More traditional

theories assume that females have a comparative advantage in home produc-

tion and males in market production, but Albanesi and Olivetti (2007) show

that improved medical capital and the introduction of the infant formula has

reduced the importance of this factor. In Becker (1985) gender differences

in earnings arise from the fact that females undertake tiring activities that

reduce work effort. So, workers with the same level of human capital, earn

wages that are inversely related to their housework commitment. The substi-

tutability between home duties and market earnings also arises in our model,

although we consider the effect of an investment in costly effort as well.

Regarding the elasticity of labor supply, Goldin (2006) documents that

the fast rise of female’s labor supply elasticity in the 1930-1970 period was the

result of a declining income effect and a rising, due to part time employment,

9Earlier models have emphasized that intrahousehold redistributional factors are impor-tant. However, these papers are either concerned with the across household heterogeneity(Apps and Rees, 2007), or follow a policy reform approach (Brett, 1998, and Apps andRees, 1999a), or focus on the positive effects of the taxation of couples (Gugl, 2004).

7

substitution effect. During the last thirty years, she argues, females started

viewing employment as a long term career rather than as a job, and this

caused a decline in the substitution effect and the labor supply elasticity.

This interpretation is consistent with how we model, in our long run setting,

the elasticity effect of a commitment to stay in the labor market in order

to take advantage of the opportunities offered by it. Blau and Khan (2007)

also document and quantify the reduction in the labor elasticity of married

women in the US, which however remains well above that of men, at a ratio

of about 4 to 1. Even in Sweden, where gender differences in labor market

outcomes are arguably less dramatic than elsewhere, Gelber (2007) estimates

that the elasticity of women is twice that of men.10

3 Exogenous Elasticities

A family consists of a male and a female who participate in market and home

activities. A costly investment in training makes a person more productive

for the market. Husband and wife share a fraction of the income they pro-

duce with market work. For the moment we let household activities in the

background and treat them as exogenous.

3.1 Setup of the Model

The index j = m, f identifies the gender. The utility function of gender j is

simply given by

Uj = Cj −1

ajL

aj

j − 1

2τ 2j (1)

where Cj is consumption, 1aj

Laj

j represents the disutility cost of supplying

Lj units of labor, aj > 2 and 12τ 2j is the cost of training. Each person is

endowed with one unit of time for work, so Lj ≤ 1. The quasi-linearity with

10Gelber’s (2007) results are also important because they analyze the responses to thevery large Swedish Tax Reform of 1991 and therefore represent a rare example of causalidentification and estimation of labor supply elasticities of household members.

8

respect to consumption allows us to obtain closed form solutions at least up

to a point. We discuss below the effect of this assumption on our numerical

results.

The timing is as follows. First, the government sets labor income taxes.

Then, the male and the female take as given the tax rates and decide indi-

vidually the amount of consumption, labor supply and training to maximize

their utilities. A perfectly competitive, constant-returns to scale firm pays

workers their marginal productivity and makes zero profits. The price of the

consumption good is one and the production function for worker j is

Qj = τjLj (2)

Therefore, the wage rate Wj equals τj. Spouse j maximizes utility taking as

given the labor income tax rate tj and the other spouse’s decisions

maxCj,Lj,τj

U = Cj −1

ajL

aj

j − 1

2τ 2j (3)

subject to

Cj = s(1 − tj)WjLj + (1 − s)(1 − tk)WkLk (4)

Wj = τj (5)

where k is the other spouse and 1/2 ≤ s ≤ 1 is the sharing parameter. Within

a single tax regime, s has the interpretation of an intrahousehold inequality

parameter. When s = 1/2, then the family fully pools its resources and

the ratio of consumption levels Cj/Ck equals 1. When s = 1, the ratio of

consumption levels is pinned down by the ratio of gross incomes,Cj

Ck=

WjLj

WkLk,

and there is no sharing of resources. We rationalize the sharing parameter as a

technological externality that captures the non-excludable and non-rivalrous,

at least to some extent, nature of the common consumption of goods within

the family.11 Finally, note that in deciding the level of training, workers

11For example, once the family purchases an electric appliance such as a refrigerator ora dishwasher it is difficult to imagine how a spouse can be excluded from its consumption.Or, the consumption of cable television from one family member does not restrict theconsumption of the good by other members of the family.

9

internalize that a higher level of investment increases their productivity and

therefore their wage rate.

The solution to the above maximization problem yields the labor supply

and the training decision functions (see the Appendix to Section 3.1 for

details)

Lj = (s(1 − tj))2

aj−2 = (s(1 − tj))2σj

1−σj (6)

τj = (s(1 − tj))aj

aj−2 = (s(1 − tj))1+σj1−σj

where

σj =∂Lj

∂Wj

Wj

Lj=

1

aj − 1(7)

is the own elasticity of labor supply with respect to an exogenous variation

in the wage rate. For this Section, cross elasticities are zero because we have

assumed quasi-linear preferences. In Section 5.4 we also discuss non zero

cross elasticities.

Suppose now that for exogenous reasons we have am > af . For the

moment we take this difference in preferences as primitive and do not explain

it as it may come from innate gender characteristics or more likely historically

induced gender roles which are especially strong in certain cultures (Alesina

and Giuliano, 2007). Under am > af the prediction of the model is that

males

• work more in the market: Lm > Lf ;

• have a lower elasticity of labor supply: σm < σf ;

• invest more in training: τm > τf ;

• receive a higher wage: Wm > Wf .

These predictions are in line with what we observe in real life labor mar-

kets. In Figures 1 and 2 we depict the labor market equilibrium. Assuming

10

that am > af , Figure 1 describes a situation in which males supply more labor

than females. This happens for two reasons. First, given an exogenous wage

rate, male participate more in the market. Second, they also invest more

in training. In turn, investment in training endogenously shifts the labor

demand curve up and increases the wage rate W . As a result the gender dif-

ferential in labor market participation and earnings expands. In Figure 2 we

describe an exogenous shift in the tax rate tj for spouse j. Taxation distorts

both the labor-consumption margin and the decision to invest in training,

so that both the labor supply and the labor demand curve shift. The final

equilibrium is characterized by lower participation in the labor market and

lower pre-tax wage rate.

3.2 Gender Based Taxation

The planner sets taxes for the male and the female in order to raise revenues

and finance a public good G. The public good does not provide utility

to anyone and the proceedings are not rebated back.12 In doing so, the

planner anticipates the private market equilibrium. Let Um(tm, tf ; am, af , s)

and Uf (tm, tf ; am, af , s) denote the indirect utility function for the male and

the female respectively. In this Section we assume that the planner weights

people uniformly, but we revisit this issue in Section 6.1 where it matters

more.13 Then, the planner solves

maxtm,tf

Ω = Um(tm, tf ; am, af , s) + Uf(tf , tm; am, af , s) (8)

subject to the constraint

tmWm(tm; am, s)Lm(tm; am, s) + tfWf (tf ; af , s)Lf (tf ; af , s) ≥ G (9)

12This is without loss in generality since the nature of the results (throughout the paper)does not change when we allow for revenues to be distributed in a lump sum way. SeeLundberg, Pollak and Wales (1997) for an natural experiment with intrahousehold lumpsum transfers.

13Under Ω = 11−e

(U1−em + U1−e

f ) with inequality aversion (e > 0), the difference in theresulting tax rates is even more profound. The same holds for the rest of the paper.

11

Proposition 1 If σm ≤ σf , then tm ≥ tf .

The proof of Proposition 1 and the intermediate derivations are presented

in the Appendix to Section 3.2. Here we just mention that (8) and (9) is

not a concave program and we have to establish sufficient conditions for the

existence of an interior global optimum with tm ≥ tf .

This proposition is an application of the Ramsey (1927) rule. It is welfare

enhancing to tax less the “commodity” which is supplied with higher elas-

ticity. If am > af , then females are more elastic and distorting their labor

and training decisions results in a greater excess burden for the society. In

Table 1 we present the welfare gains when moving from a single tax to dif-

ferentiated taxes by gender. Gender Based Taxation (GBT) generates more

equality in labor market outcomes. For conservative values of the elasticity

ratio such as σm

σf= 1

2, which was recently estimated by Gelber (2007) for

Sweden, GBT raises welfare and GDP by more than 1%. For an elasticity

ratio of σm

σf= 1

3the gains from GBT exceed the 4% of GDP.14 Naturally,

GBT is more efficient the higher is the level of distortions (i.e. the higher is

public expenditure G) and the lower is the ratio of elasticities σm

σf. We defer

the discussion of how the resource sharing parameter s affects the gender

taxes for Section 6.2.

These GDP gains are very large, possibly unreasonably so. We also have

explored other examples which eliminate the quasi-linearity with respect to

consumption. For reasonable parameters and functional forms we find that

with a ratio of elasticities σm

σf= 1

4(as in Blau and Kahn (2007) for the US),

GBT raises GDP by approximately 1.24%.15

14For cross-country evidence on the gender differential on labor supply elasticities seeAlesina, Glaeser and Sacerdote (2005), Blau and Kahn (2007), Blundell and MacCurdy(1999).

15For this exercise we use the standard CRRA/power expression for the subutility ofconsumption which induces both substitution and income effects on labor supply. In thiscase, the concavity of the subutility function for consumption mutes the welfare gains fromthe reallocation of resources. For more details see the Appendix to Section 3.2.

12

4 The Organization of the Family

Thus far we have assumed that different labor market behavior of men and

women derive from exogenous differences in preferences and attitudes. That

is, we have taken the key parameters am and af as our primitives. In what

follows we propose a possible formalization of the household allocation of

home duties which derives these parameters endogenously.

A family has to undertake 2A family duties, or chores. Each duty is per-

formed by one spouse. When a spouse performs one home duty she/he gets

nothing while the other spouse gets a positive shock in the labor market. The

argument is similar to that of Becker (1985) who posits that the spouse who

does more homework has fewer “energy units” to allocate into the market.

Therefore, there are 2A corresponding labor market shocks that hit the

family. The shocks are assumed to be i.i.d. and denoted as xi. Each random

variable xi is distributed as a chi-squared with one degree of freedom, i.e.

xi ∼ χ21. Let 2am be the number of xi shocks that the male absorbs; each

shock corresponds to one unit ”off-duty” that he gets. 2af = 2(A − am) is

the amount of home duties that the male gets, and therefore it is also the

number of labor market shocks that the female absorbs. By the properties

of the χ2 distribution we can define an ”aggregate shock” for the male as

ωm =∑2am

i=1 xi, with support in [0,∞) and expected value E(ωm) = 2am.

Similarly for the female we have that ωf =∑2A

i=2am+1 xi, with support in

[0,∞) and E(ωf ) = 2af . Ex post utility for spouse j = m, f is defined over

bundles of consumption, labor and training and given by

Vj = Cj −1

ajev(Lj)ωj − 1

2τ 2j (10)

where C is consumption, L is labor supply in the market, and τ is amount

of training. The subutility of labor is given by v(Lj) = 12

(1 − 1

Lj

)< 0,

with v′ > 0, v′′ < 0 and Lj < 1. This specific ”χ2 shocks - CARA utility”

environment is adopted to obtain the more familiar CRRA representation of

13

the (ex ante) utility function that we used in Section 3.

To fix ideas about the nature of the shocks, consider the situation where

the male and the female decide how to allocate home duties over a period

of two weeks. Specifically, for each weekday, one of the two spouses must be

in “charge of the kids” (i.e. take them to school, make sure that they have

their time after school organized etc.).16 This hypothetical situation can be

mapped in our notation as follows. 2A = 10 is the total number of days in

which one parent has to take the kids to school while the other is exempted

from these home duties. 2am is the number of days that the male is not in

charge of the kids and therefore 2af is the total number of days where the

male is in charge of the kids. For each of the i = 1, ..., 2am days where the

father is not in charge of the kids and works in the market, there is a positive

shock xi that affects his utility of working in the market. To put it differently

(and with a slight abuse of language), in the days in which a spouse is not

in charge of kids, she/he has more energy and can make “things happen”

at work and get a positive utility reward. There are also days in which the

spouse is in charge of the children and work provides only the basic wage

with no upside options.17

The expost utility of working in the market for spouse j is given by the

term − 1aj

ev(Lj)ωj< 0. Given a realization of ωj, a higher amount of labor

supply decreases utility. For given amount of labor supply, a favorable real-

ization of ωj increases the utility of working in the market (or decreases the

disutility of working). Since the shock ωj has not been realized when spouses

decide how much to consume, supply labor and invest in costly training, we

need to work with the ex ante utility function. Using the moment generating

function of a chi-squared random variable with 2aj degrees of freedom we

16In this sense one cannot ”quit a child” and home duty in our model is intrinsicallydifferent from having a second job.

17The abuse of language is that we do not model energy explicitly. Instead, taking lesshome duties directly implies the possibility or receiving a higher labor market shock.

14

obtain 18

Uj = EωjVj = Cj −1

aj

Laj

j − 1

2τ 2j (11)

The “χ2-CARA” expost representation of preferences in (10) allows us

to work with the familiar CRRA-power expression for labor supply in (11),

which is the utility function used in Section 3. With this derivation we

intend to provide a rationale for the key parameter aj. While in the previous

section gender differences were “innately” built in preferences (so that am and

af were “genetically” or “culturally” fixed in a permanent way), in Section

5 we develop a bargaining game which delivers the equlibrium division of

chores between the two spouses and ultimately determines endogenously their

market participation and elasticity.

The expected marginal utility of working is given by

ULj = −Laj−1j with aj > 2 and Lj < 1 (12)

so that fewer home duties (higher aj) increase the expected marginal utility

of working for spouse j. Because the latter expects a higher realization of the

labor market shock ωj , he or she works more, invests more in training and

earns a higher wage rate. This means that home duties and participation in

the market are substitutes. At the same time, assuming fewer home duties

implies a higher elasticity of the expected marginal utility of working with

respect to labor supply

εULj,Lj =

ULLL

UL= (aj − 1) =

1

σj(13)

Since the lower the amount of home duties the more sensitive is the marginal

utility of working to movements in the supply of labor, a given change in the

wage rate Wj meets with a smaller movement in labor supply Lj in order to

restore the first order condition for labor supply. This implies that spouse j

has a less elastic labor supply.

18We have that for a random variable ωj ∼ χ22aj

the moment generating function eval-

uated at some q < 1/2 is given by Mω (q) = Eω (eqωj ) =(

11−2q

)aj

.

15

Thus, the gender gap in labor supply elasticities can be traced back to the

attitudes of the two spouses towards risk and to the differences in the access

to labor market shocks which is determined by the bargained allocation of

home duties. For spouse j and given a specific realization of the labor market

shock ω, we define u = − 1aev(L)ω to be the expost disutility from labor supply.

We also define the curvature functions rω = −uωω

uωand rL = uLL

uLas measures

of the attitude towards risky realizations of ω and L respectively. Then we

can show that 19

∂rω

∂L= −∂rL

∂ω= −v′(L) < 0 (14)

The first part of the symmetry condition (14) states that a spouse who

participates more in the labor market is less risk averse to stochastic real-

izations of ω. This third-order cross partial effect is a diversification motive.

High realizations of L cause spouse j to be less averse to ω-uncertainty since

uncertainty “per unit” of labor decreases. The second part of equation (14)

states that a spouse getting a good realization of ω is more risk averse to

stochastic realizations of participating in the market L.

The intuition for the gender gap in elasticities is that if men get fewer

home duties (am > af), then they get a higher number of positive shocks to

the utility of working. For men, this expectation of more favorable labor mar-

ket opportunities (a high realization of ωm) is an expected intrinsic benefit

from working. So men prefer to commit to a larger amount of labor ex ante,

which is also more stable because it is calibrated not only on the wage but

also on the intrinsic expected benefit of working. When the wage changes,

this committment makes them less willing to adjust their labor supply. The

reverse happens for women.

19We don’t have a minus sign in the definition of rL because labor is a “dis-commodity”,i.e. uL < 0. See Appendix to Section 4 for more details.

16

5 Household Bargaining

5.1 Timing

The timing is as follows. First the government sets the tax rate(s). Then

the family members bargain over the allocation of home duties, which results

in equilibrium values for aj. Next, labor supply decisions are taken, wages

paid, shocks realized and consumption shared. A commitment technology

makes it impossible for the government to change the tax rates after family

bargaining decisions are made or after the realization of labor market shocks.

5.2 Bargaining over Home Duties

At the second stage of the game the couple decides whether to marry or

not. If the male and the female decide to marry, then they bargain over the

allocation of home duties, A = am + af . In doing so, they both rationally

anticipate the resulting labor market equilibrium. The utility of a spouse j

when married is given by the indirect utility function at stage 3, as described

by the maximization of (3) subject to the constraints (4) and (5). We assume

that the autarky utility level of each spouse (the threat point), is given by

the value function of the following program

maxCj,Lj ,τj

Tj = Cj −1

φLφ

j − 1

2τ 2j − z (15)

subject to

Cj = (1 − tj)WjLj and Wj = τj (16)

The disutility of being alone is z . A single does not share resources so he

or she gets a “full share of a smaller pie”. Importantly, a single has a shock

ωs ∼ χ22φ with φ = A, which means that singles take less home duties than

a married person, for instance because they have no children.20 Translated

20This assumption can be relaxed. Even when a single has the same amount of homeduty as a married person on the equilibrium path, the results do not change. See theAppendix to Section 5.3.

17

into the words of the example in Section 4, a single person never has to drive

the kids to school.

Given this specification of the utilities in marriage and in autarky, for

any pair of taxes (tm, tf), the maximization of the asymmetric Nash-product

delivers the allocation of home duties:

[Um(am; tm, tf , s)− Tm(tm, φ, z)]γ [Uf (am; tf , tm, s)− Tf (tf , φ, z)]1−γ (17)

where γ is the bargaining power of the husband.

We study the case of γ > 1/2, which can be justified as the historical in-

heritance from a time in which physical power mattered, with cultural forces

persistently affecting family formation.21 The fact that men have stronger

bargaining power, seems consistent with survey evidence. Friedberg and

Webb (2006) use data from the Health and Retirement Study and document

that nearly 31% of males believe that “they have the final say in major deci-

sions” while only 12% believe that their spouse is in the same condition. At

the same time, approximately 31% of the females admit that their husband

has the final say while only 16% believe to have the final say in major deci-

sions. A biased allocation of home duties in favor of the male accords well

with the existent empirical evidence. Aguiar and Hurst (2007) show that

although the difference between male and female house work has decreased

during the last 50 years, females still perform nearly twice as much homework

as males.

Our marriage specification is, admittedly, simplified, but note that it

assumes that only the explicit bargaining power γ is exogenous, while the

effective implicit bargaining power, that derives from the combined effects of

γ and the threat points, is endogenous and indeed depends on GBT. There

21The effects and causes of different family structures with specific reference to the roleof women and allocation of home duties has been the subject of empirical cross countryresearch by Alesina and Giuliano (2007), and Fernandez (2007). Their results suggestthat one should be cautious in applying to different countries and cultures the same set ofpreferences on the issue of gender roles.

18

is a feedback effect from government policy to the intra-household allocation

of bargaining power because the outside option of a spouse j depends on the

tax rate tj. For example when the tax rate decreases, spouse j acquires more

implicit bargaining power through increased training, wage rate and market

participation.22

5.3 Properties of the Bargaining Solution

We consider the properties of the solution mapping am(tm, tf) : [0, 1]2 7→(2, A−2). The bargaining solution prescribes how the family allocates home

duties for any pair of tax rates, given parameters γ, s, A and z. We cannot

derive closed-form expressions for the solution am(tm, tf) and its compara-

tive statics, but we can discuss intuitively (and establish numerically) two

important properties of the bargaining solution. For more details see the

Appendix to Section 5.3.

First, the sharing parameter affects the allocation of home duties. Specif-

ically for given (tm, tf), an increase in s, i.e. less resource sharing, makes the

allocation of shocks more unbalanced, ∂am

∂s> 0. We call this the sharing

effect and depict it in Figure 3. This Figure plots the male’s indirect util-

ity function Um(am) as a function of home duties. When the male makes

take it or leave it offers to the female (equivalent to γ = 1) and there is

no income sharing, he chooses the maximum feasible level of am, that is he

chooses not to take any home duties. As the sharing of resources becomes

important (s decreases) the male decides to take some amount of homework,

even though he has the maximum level of bargaining power. When income

is shared, it is never individually optimal for the male to have the female not

22Pollak (2007) argues convincingly that the wage rate and implicitly the level of humancapital should determine the outside option of a spouse. Our specification addresses, atleast partly, this concern because taxes distort the training decision and endogenously shiftthe labor demand curve. However, as we discuss in the concluding section, our specificationdoes not recognise dynamic elements of acquiring bargaining power such as investment inhuman capital.

19

working in the market. The same intuition applies for any level of bargaining

weights (i.e. 1/2 < γ < 1). As income pooling becomes more important the

intra-household allocation process becomes more balanced. At some level sE,

resource sharing is so important that the allocation of shocks is completely

balanced, even without GBT.

The sharing of resources implies that there is an externality in the model.

Inspection of the solution (6) suggests that as resource sharing increases,

both spouses participate less in market activities because they lose part of

their individual claims over the market product.23 For extreme levels of

sharing, s < sE, the male with the bargaining power is better off by staying

at home and having the female working and sharing her income with him.

This prediction is not realistic and from now on we restrict attention to

s ≥ sE.24

By increasing the tax rate for the male tm and keeping fixed the female’s

tax rate tf and the level of sharing s we can examine the second property of

the bargaining solution. Three are the relevant effects:

• Redistribution Effect : ∂Um

∂tm< 0. When tm increases, the male is worse

off inside the marriage and demands a lower amount of home duties

(higher am) in order to “stay in the contract”.

• Threat Effect : ∂Tm

∂tm< 0. When tm increases, the male is worse off

outside the marriage and his implicit bargaining power decreases. This

means that he is willing to accept a higher amount of home duties

(lower am) in order to “stay in the contract”.

23In that sense this model is a little more individualistic that the collective family modelof Chiappori (1988, 1992).

24Even though, for given allocation of home duties, a spouse works and invests less thegreater is the sharing of resources, the intra-household allocation process in the bargainingstage of the game is always efficient because the Nash bargaining process is Paretian.Referring to Figure 5, note that the allocation of resources always lies on the Paretofrontier because we cannot make one spouse better off without worsening the position ofthe other (see also equation (20)).

20

• Cross Redistribution Effect :∂Uf

∂tm< 0. Because spouses share resources

inside the marriage, a higher tm makes the female worse off inside the

marriage. In order to “stay in the contract” she must be compensated

with less home duties (lower am).

We can show (see the Appendix to Section 5.3) that the threat effect

always dominates the redistribution effect. That is, a higher tax rate brings

a more balanced allocation, ∂am

∂tm< 0 because 25

∂Um

∂tm− ∂Tm

∂tm> 0 (18)

which holds if (but not only if) s < 1 and am < φ. Similar reasoning (but

not symmetric because γ > 1/2) holds for varying the female’s tax rate and

∂am

∂tf> 0.

We sum up this discussion in Figure 4 which depicts the solution to the

bargaining program as a function of the sharing of resources s and the ratio

of taxes tmtf

.

5.4 Cross Elasticities

With an endogenous allocation of home duties the cross elasticities of labor

supply are not zero as in Section 3. We can write for spouse k

eLk,tj =∂Lk

∂tj

tj

Lk

=

(∂Lk(ak)

∂tj

+∂Lk

∂ak

∂ak

∂tj

)tj

Lk

(19)

The term ∂Lk(ak)∂tj

in (19) is the response of k’s labor supply to j’s tax rate

for a given allocation of home duties. This is zero as in Section 3 because

preferences are quasi-linear and the budget constraint is separable in spouses’

net incomes. The term ∂Lk

∂ak

∂ak

∂tjappears because the allocation of home duties

is endogenous and responds to variations in the tax rate. For instance, a

25See the Appendix to Section 5.3 for the robustness of this result after considering thesecond order effects.

21

higher tax rate for the male tm, increases the relative bargaining power of

the female. As a result, the female takes less home duties (af increases), and

the cross elasticity of labor supply with respect to her spouse’s tax rate is

positive.26

6 Gender Based Taxation

6.1 Government Objectives

As we discussed in Section 3, GBT is optimal if the labor supply of men is less

elastic than than that of women. Based on our derivation in Section 4 this

happens when males assume fewer home duties than women and women have

no comparative advantages in home duties. Obvioulsy then with γ = 1/2,

the market and non-market behavior between spouses is identical and there

is no need for GBT. However, as discussed before, there is ample evidence

for gender differences in labor market participation rates and elasticities and

for a biased allocation of home duties and decision making power within

the family, suggesting that γ must be greater than 1/2 for our model to be

a good description of the real world. But in a world where γ > 1/2 how

should a social planner evaluate the utility of men and women? A natural

premise is that the social planner evaluates people equally, that is we adopt

the utilitarian welfare function, Ω = Um + Uf . Thus we have a sort of

“social dissonance” (Apps and Rees 1988) between the preferences of society

(as for example implied by the utilitarian function Ω) and the equilibrium

result of an intrafamily game in which one party has a disproportionate share

of power. If this the case, then, there is a “distortion” that could justify

government’s intervention which, in addition to financing the public good

efficiently according to the Ramsey rule, affects intra-family bargaining as

26This is in line with the empirical evidence, see for example Aaberge and Colombino(2006) for negative cross wage elasticities in Norway and Blau and Kahn (2007) for theUS.

22

well.

If the government could choose directly the allocations of home duties and

then set taxes to raise a pre-specified amount of revenues, it would choose

the ungendered equilibrium (am = af , tm = tf) and there would be no need

for GBT; this would be the first best. In Figure 5, we depict this Edge-

worth’s (1897) “egalitarian” solution: starting from a gendered equilibrium

(am > af), we can allocate one more unit of home duty to the male from the

female and increase social welfare because there are “decreasing returns to

specialization”. In other words, the first hour that the father spends with his

children is more productive than the female’s last hour. This is true because

starting from am > af we have

∂Ω

∂am

=∂Um

∂am

+∂Uf

∂am

< 0 (20)

Although the government cannot dictatorially impose a balanced intrahouse-

hold allocation of duties, gender based taxes affect the allocation of chores

bewteen spouses and can bring the society closer to the first best.

6.2 The Organization of the Family

The planning program is

maxtm,tf

Ω = Um(tm, tf ; am, af , s) + Uf(tf , tm; af , am, s) (21)

subject to the constraint

tmWmLm + tfWfLf ≥ G (22)

The difference with respect to Section 3.2 is that now the allocation of

home duties is endogenous and the government anticipates it. That is:

am = am(tm, tf ; γ, s, z)

af = af(tf , tm; γ, s, z).

23

Wj = Wj(tj, aj(tj, tf))

Lj = Lj(tj, aj(tj, tf))

for j = m, f .

Starting from a single tax rate, the government can induce a more bal-

anced allocation by differentiating taxes and setting tm > tf . As long as

labor supply elasticities remain different (σf > σm), GBT also reduces fiscal

distortions as in Section 3.2. There is an implicit cost, however, of taxing

the male at a higher rate: not only we distort his labor supply and training

decisions (as in Section 3.2) but also we force him endogenously to take more

home duties (lower am) which further reduces the government’s ability to

extract revenues from the primary earner. This “Laffer curve” effect appears

in the first order conditions and increases the ratio of the female over the

male marginal revenue (see Appendix to Section 6.2 for further elaboration).

It can be inspected by looking at the bliss point of spouse j under exogenous

and endogenous bargaining. For the former case, the peak of the Laffer curve

is given at the point where the elasticity of earnings with respect to the tax

rate equals -1

tbj =

Ej

−dEj

dtj

=aj − 2

2aj

=1 − σj

2(1 + σj)(23)

where Ej = WjLj are pre-tax earnings. Notice that if a higher tj reduces aj,

then the peak of the Laffer curve shifts to the left. Then, for the endogenous

bargaining case we have that

tbj =

Ej

−∂Ej

∂tj− ∂Ej

∂aj

∂aj

∂tj

(24)

with tbj < tb

j as long as∂Ej

∂aj> 0 and

∂aj

∂tj< 0 as it is the case for the male.

In Figure 6 we depict the solution for the ratio of optimal gender based

taxes tmtf

as a function of the sharing parameter s.27 There are three areas of

interest. In Area I the externality is so high (s < sE) that the male decides to

27See Appendix to Section 6 for details on the calibration and the solution.

24

stay at home. The female works more, earns more, is less elastic and the male

enjoys resources mainly from his spouse’s income. As mentioned before, this

case does not accord with real life labor markets and we can safely dismiss

it.28 In Area II, the male has the bargaining power and without extreme

sharing of resources he prefers to assume fewer home duties. As a result he

works, invests and earns more than the female. The analysis of Section 4

applies, so the male is also less elastic. In Figure 6 we depict the ratio of

labor supply elasticities (that move in the opposite direction of the ratio of

home duties) under a single and gender based taxes together with the ratio

of optimal taxes, tmtf

. Gender based taxes induce a more balanced allocation

of home duties and bring closer to 1 the ratio of elasticities because they

increase the implicit bargaining power of the female. Moreover, as long

as σf > σm, the conventional Ramsey principle applies and GBT reduces

fiscal distortions. Note that with endogenous bargaining and starting from

γ > 1/2, it is relatively more costly for society to tax females than it is

in the exogenous bargaining case. The reason is that every extra unit of

tax revenues that the government raises from the female further deteriorates

her implicit bargaining power and results in a more gendered allocation (see

Appendix to Section 6.2 for this argument).

In Area III, tm > tf is still optimal. In this region, with less resource

sharing and given the intuition of the sharing effect in Section 5.3, the ratio

of home duties and the ratio of elasticities diverge even more. However, the

ratio of tax rates starts to decline. The intuition for this result is given in

Figure 7. This Figure depicts the ratio of wage rates (or training levels) Wm

Wfas

a function of the sharing parameter s under a single and gender based taxes.

Note that the ratio of optimal taxes tmtf

in Figure 6 traces the interhousehold

inequality I = Wm

Wfthat prevails under a single tax rate in Figure 7. The

reason is that GBT reacts to the male over female wage ratio under a single

28See also the references for the empirical failure of the income pooling hypothesis givenin Section 2.

25

tax rate, which is a measure for the misallocation of home duties without

government intervention.29 At the same time, GBT targets the wage ratio

under differentiated taxes, because this is correlated with spouses’ relative

decision making power.30 Since GBT reallocates efficiently the bargaining

power between spouses, the ratio of wage rates under GBT, and therefore

the relative decision power, shifts down relative to the single tax rate, as

shown in Figure 7. At some level of sharing sM , however, the household by

its own reduces inequality in earnings and since it is costly for the government

to further increase the elasticity of the primary earner, there is no reason why

the ratio of taxes should continue to diverge for s > sM = .92.

The reason why resource sharing and inequality under a single tax rate

exhibit an inverted U-shaped relationship is the following. Under a high

level of resource sharing, both the male and the female participate less in

the market, and the inequality ratio is low. As resource sharing declines (s

increases) both partners participate more, but the male at an increasing rate

and therefore inequality starts to rise. Under extremely low levels of income

pooling the female starts to participate at an increasing rate, so inequality

begins to fall. Even for no resource sharing, i.e. s = 1, we always have

Wm > Wf , so the government would always set tm > tf . See Appendix to

Section 6.2 for more details.

In Figure 8 we depict the gains in welfare, GDP and employment when

moving from a single to gender based taxes as a function of s. The gains

are highest when pre-gender based inequality is at its maximum and begin

to fall when GBT becomes less beneficial as in Area III.

Finally, in Figure 9 we depict the possibility that both spouses gain under

GBT. If resource sharing is important, both spouses gain when moving from

a single to gender based taxes because the female starts to work, train and

29Under a single tax rate the ratio of wages Wm

Wfis higher the more gendered is the

allocation of home duties. See equation (29) in the Appendix to Section 6.2.30In particular, we can write autarky utilities as Tj = φ−2

2φ W 2j (tj) − z.

26

earn more, a decision which is not internalized by the family when spouses

individually decide how much to participate in the market.

7 Conclusions

In this paper we begin to analyze the effects of Gender Based Taxation as

a potential tax policy. If the intra-family bargaining process favors the hus-

band, GBT with lower tax rates for females is superior to an ungendered

tax rate. In what one could label the “short run”, namely before the fam-

ily organization adjusts to the new tax regime, GBT reduces tax distortion

because of the Ramsey principle according to which one should tax less com-

modities with higher supply elasticities. When the spouses react to GBT by

re-bargaining over household duties, GBT leads to a more equitable distri-

bution of household chores and market activities. To the extent that this

reallocation does not produce complete equity between male and female and

therefore the supply elasticities remain different, GBT is optimal. The real-

location towards more equality of household duties is an additional welfare

improving effect if society evaluates the welfare of males and females equally.

In the “long run”, the welfare gains of GBT derive both from the Ramsey

principle and from a more “efficient” organization of the family that takes

into account the decreasing marginal benefits in home versus market ac-

tivities. In our model GBT is optimal for the couple with both members

weighted equally and, for some parameter values, for both members of the

couple individually.

Rather than reviewing in more details our results it is worth discussing

several important avenues for future research. First, we have not considered

the possibility of a comparative advantage of females in home production.

Albanesi and Olivetti (2007) point out that technological improvements have

certainly reduced women’s comparative advantage in household production

27

and duties, nevertheless comparative advantage may still exist.31 In this

case there would be forces going in opposite directions. On the one side, the

government does not want to impose lower taxes on women and encourage

female market participation because this would oppose possible increasing

returns that the household enjoys when spouses specialize in market and

non-market activities. On the other side, imposing higher taxes on females,

as discussed in Section 5.3, results in a further deterioration of their implicit

bargaining power and opens up the gap in the labor supply elasticities. When

elasticities diverge, we expect the Ramsey effect to become stronger and

counterbalance the effect of comparative advantage. Which of these two

effects prevails is an open issue that requires more theoretical and empirical

work.

Second, our model does not allow for a realistic marriage market since

it considers a society in which marriage is optimal for everybody along the

equilibrium path. A proper discussion of the marriage market would require

the introduction of some heterogeneity within the pool of men and women

and the consideration of a matching or a searching model.

Third, in the present model the word “training” can be interchanged

with“effort”. The training decision is taken when the couple is already

formed. Therefore, we cannot analyze a situation in which a man or a

woman, when unmarried, invest in training as a commitment to gain bar-

gaining power. This interesting extension could be discussed in an even more

general model in which the marriage market is also endogenized. A key ques-

tion that this analysis could help answering is whether or not GBT should

refer to only married couples or to males and females regardless of their

marriage status. Alternatively, if we allowed for different tax rates not only

across genders but also within genders, our model would suggest that taxing

31Note that Ichino and Moretti (2006) find that biological differences explain a largepart of the gender differential in absenteeism which translates in a 12% fraction of theearning gap.

28

single men at a higher rate might be a superior policy because it reduces di-

rectly the autarky utility of men, inducing them to accept more home duties

in order to marry. An evaluation of these more complicated tax structures

would depend undoubtedly on their redistributive properties in a world of

heterogeneous households.

Fourth, our model does not distinguish between the intensive and ex-

tensive margins of labor supply decisions. There is instead an important

discontinuity between starting to work from inactivity and increasing work-

ing time if someone is already active in the market.

Finally, we believe that a comparison of Gender Based Taxation with

other gender and family policies, such as quotas, affirmative action, forced

parental leave and public supply of services to the families, is necessary within

a unified theoretical framework in order to draw policy conclusions. We see

no reason why GBT should not be an excellent “horse” in a race with all

these alternative policies. In fact our basic economic intuition regarding

the superiority of price incentives versus quantity restrictions or regulations

would make GBT a favorite in the race, but we still have to run it.

29

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[41] Mirrlees, J. A., (1971), ”An Exploration in the Theory of OptimumIncome Taxation”, Review of Economic Studies, 38, 175-208.

[42] Myles, G., (1995), Public Economics, Cambridge University Press.

[43] Pollak, R., (2007), ”Family Bargaining and Taxes: A Prolegomenon tothe Analysis of Joint Taxation”, IZA Working Paper No. 3109.

[44] Piggott, J. and J. Whalley, (1996), ”The Tax Unit and Household Pro-duction”, Journal of Political Economy, 104, 398-418.

[45] Ramsey, F. P., (1927), ”A Contribution to the Theory of Taxation”,Economic Journal, 37, 47-61.

[46] Rosen, H., (1977), ”Is It Time to Abandon Joint Filing?” National TaxJournal, 30, 423-428.

[47] Samuelson, P., (1956), ”Social Indifference Curves”, Quarterly Journalof Economics, 70, 1-22.

[48] Thomas, D., (1990), ”Intra-household Resource Allocation: An Inferen-tial Approach”, Journal of Human Resources, 25, 635-664.

[49] Vermeulen, F., (2002), ”Collective Household Models: Principles andMain Results”, Journal of Economic Surveys, 16, 4, 533-564.

33

Appendix

Appendix to Section 3.1

Equation (6) is derived by substituting constraints (4) and (5) into the objectivefunction (3), taking the first order conditions with respect to L and τ and solvingthe resulting system of equations. The second order sufficient conditions for thismaximization problem hold, as ULL < 0, Uττ < 0, ULLUττ −U2

Lτ > 0 (U is globallystrictly concave in (L, τ)). Equation (7) gives the elasticity of labor supply withrespect to an exogenous variation in the wage rate. This is derived from thefirst order condition with respect to labor supply (for given amount of training),s(1− tj)Wj −L

aj−1j = 0. Starting from am > af , the comparative statics on wages,

labor supply, elasticities and training, follow directly from inspection of (6) and(7).

Appendix to Section 3.2

We first substitute the solution (6) and the constraints (4) and (5) into the ob-jective function (3) and derive the indirect utility function for gender j. Wedenote the revenues collected from gender j evaluated at the solution (6) asRj = tjLj(tj ; aj, s)Wj(tj ; aj , s). Then we can write the planning program asmaxtm,tf Ω = Um(tm, tf ; a, s) + Uf (tf , tm; a, s) subject to R = Rm + Rf ≥ G.

A standard complication in public economics (Diamond and Mirrlees 1971b,Myles 1995, pp 113-114) arises from the fact that the above maximization problemis not sufficiently ”concave”. In the dual approach to public finance, the problemarises because the consumer’s indirect utility function is quasi-convex in prices(and income). The program can be turned into a concave problem for a socialwelfare function of the form Ω(Um, Uf), with Ω being sufficiently concave (highinequality aversion), but in general the transformation of a concave with a convexfunction is not guaranteed to be concave. In our case, with quasi-linear preferencesand the utilitarian welfare (i.e. ΩUm = ΩUf

= 1), welfare is strictly convex in thetax rates. This means that we cannot invoke standard sufficient conditions fromthe theory of concave programming.

To establish the existence of the solution and the sufficiency of the first ordercondition for the above problem we follow fairly standard steps.32 First, from thedefinition of the indirect utilities Um and Uf , it is straighforward to show that thewelfare function Ω = Um + Uf is strictly decreasing in tm, strictly decreasing in

32We also can show that this is true for any welfare function that is more concavethan the utilitarian case (which is the least concave welfare function).

34

tf and strictly convex in (tm, tf).33 So, in the (tm, tf ) space, which is depicted inFigure 10, the gradient vector ∇Ω points towards the origin (0, 0) and the lowercontour set of the social indifference curve Ω(tm, tf ) = Ω is strictly convex.

Second, consider the revenue function for spouse j, Rj . We have that ∂Rj

∂tj=

sL2j [1−tj−tj(

aj+2aj−2)]. The Laffer curve for spouse j peaks at the tax rate where the

elasticity of earnings with respect to the tax rate is minus unity, so that tbj = aj−22aj

.

We also have that ∂2Rj

∂t2j= sL2

j [−1− aj+2aj−2 ]+2sLj

∂Lj

∂tj[1−tj−tj(

aj+2aj−2)]. The first term

is negative while the second term is negative if ∂Rj

∂tj> 0. So, the revenue function

for spouse j is concave if (but not only if) we are at the upwards sloping part ofthe Laffer curve. Given the properties of Rm and Rf , total revenues R = Rm +Rf

are strictly increasing in each of tm and tf and strictly concave in (tm, tf) if (butnot only if) (tm, tf) < (tbm, tbf). This means that in the (tm, tf) space the gradientvector of the revenue function ∇R points towards the bliss point and the uppercontour set of the revenue isolevel R = G is strictly convex in that region. Notethat if am > af , then the bliss point lies above the 45 degree line which signalsthat tm > tf holds in the solution. So, if the government wants to extract themaximum revenue the solution is tm = tbm > tf = tbf . We define Gmax to be themaximum sustainable level of public expenditure with Gmax = R(tbm, tbf).

Next, it is easy to show that tm > tbm or tf > tbf can never solve the program.If this was not the case, then we could increase both welfare and revenues whichcontradicts optimality. Therefore, without loss in generality we now restrict at-tention to the set D = [(tm, tf) : tm ∈ [0, tbm], tf ∈ [0, tbf ], Gmax ≥ R ≥ G]. SinceD is a compact set and Ω is a continuous function, by Weierstrass Theorem, aglobal maximum exists in D. Finally, it must be the case the constraint alwaysbinds at the optimum. If this was not the case, then we could increase welfare bydecreasing some tax rate, while still satisfying the constraint.

Fix an arbitrary level of public expenditure. Since we know that tm > tf ifG = Gmax we now restrict to G < Gmax. As we showed before, in this area, therevenue function is strictly increasing and strictly concave. The next step is toestablish that for am > af , i.e. σf > σm, tf > tm is never an optimal solution.To show that this cannot be an optimum, it suffices to show that the slope of thewelfare function in the (tm, tf) space is always greater in absolute value than theslope of the revenue function at every point along the R = G level where tf > tm.This is shown in Figure 10. The reason is that then we can decrease tf , increase tm,increase welfare and still satisfy the budget constraint. tf > tm is never optimalbecause the relative marginal cost of taxing a female is higher than the relative

33Because labor supply and training are always in the interior all inequalities arestrict.

35

marginal revenue that the government extracts.

The slope of the revenue function is given by −Rtf

Rtm= −

sL2f [1−tf−tf (

af +2

af−2)]

sL2m[1−tm−tm(am+2

am−2)]

and

the slope of the welfare indifference curve by −Ωtf

Ωtm= −

sL2f (1−tf )[s+(1−s)(

2afaf−2

)]

sL2m(1−tm)[s+(1−s)( 2am

am−2)].

Now starting from am > af (σf > σm) and tf > tm we have thats+(1−s)(

2afaf−2

)

s+(1−s)( 2amam−2

)is

larger than one larger than1−

tf1−tf

(af+2

af−2)

1− tm1−tm

(am+2am−2

)for all s, so that the welfare indifference

curve is steeper than the revenue level at any point where tf > tm holds.Similarly, we can establish that the only point where we cannot increase welfare

without violating the constraint is the tangency point (notice however that we hadto go through this argument first). In that point the welfare indifference curve isless convex than the budget constraint and the optimal taxes satisfy the condition

s + (1 − s)( 2af

af−2)

s + (1− s)( 2amam−2)

=1 − tf

1−tf(af+2

af−2)

1− tm1−tm

(am+2am−2)

(25)

Equation (25) establishes that if σf > σm then tm > tf . The tangency condi-tion is unique because the utility function is strictly concave. This ensures that theobjective function (8) is strictly convex and the constraint (9) is strictly concavein the tax rates.

For the version of the model without quasi-linear preferences, the utility func-

tion for spouse j is given by Uj = Xj − 1aj

Laj − τ2j

2 , where Xj is a ”composite

commodity” given by Xj = sC1−θ

j

1−θ + (1 − s)C1−θk

1−θ and the budget constraint issimply Cj = (1 − tj)WjLj . In Table 1, we set G

GDP = 20% and θ = 0.5.

Appendix to Section 4

The term 1a in front of the disutility of labor in (10) is just a normalizing constant

and it is easy to verify that cross gender differences hold as before even withoutthis constant. U = EωV = C − 1

aLa − 12τ2 is the expected utility function which

is derived under the properties of the chi-squared distribution. Then, equations(12)-(13) are obvious. u = − 1

aev(L)ω is the expost disutility from labor supply (inV = C + u − 1

2τ2), where v(L) = 12(1 − 1

L), with v′ > 0, v′′ < 0, v′′′ > 0. We alsodefine the curvature functions rω = −uωω

uω= −v(L) and rL = uLL

uL= v′′(L)

v′(L) +v′(L)ω.While we have that rω > 0, so spouses are always risk averse in ω-variations, rL > 0only for ω > 4L. However, every spouse expects to be expost averse to L-variationsbecause EωrL = v′′(L)

v′(L) + 2av′(L) > 0. rω is constant in ω (hence the terminology

36

”CARA”) but changes with L. rL is not constant but depends on L and ω. Forthe third order effect ∂rL

∂L , a sufficient, but not necessary, condition for this to benegative (”risk prudence”) is that aj > 3.

What matters for our results is the, in expectation, variation of rL with theexpected opportunity offered by the labor market, E(ω). More specifically, EωrL

is positively correlated with the E(ω) because

∂Eω(rL)∂E(ω)

=1

2L2> 0 (26)

(26) states that as the expectation of favorable labor market opportunities in-creases, a spouse expects to be less willing to adjust labor supply L expost. Thisflattens the (exante) utility contours (in the (C, L) space) and lowers the elasticityof labor supply.

Appendix to Section 5.3

Denoting a = am, the derivative of the indirect utility function for the male withrespect to a is given by ∂Um

∂a = 1aLa[ 1

a−ln L]+(1−s)(1−tf )∂Ef

∂a , where Ef = WfLf

are female’s earnings. In the absence of sharing we have ∂Um∂a > 0. In the presence

of sharing, the second term tends to lower ∂Um∂a because the male loses consumption

by forcing the female to stay at home. For extreme levels of sharing, ∂Um∂a < 0.

This is Area I in Figure 6. Similarly for the female. From now on we restrict thediscussion in Areas II and III, with ∂Um

∂a > 0 and ∂Uf

∂a < 0. In all Pareto efficientallocations, ∂Um

∂a and ∂Uf

∂a have the opposite sign.Write the first order condition for the maximization of (17) as F (a, tm, tf , s, γ) =

γ∂Um(.)

∂aUm(.)−Tm(.) + (1 − γ)

∂Uf (.)

∂aUf (.)−Tf(.) = 0. Differentiating this identity with respect to

a we get that −∂F∂a = γ[∂Um\∂a

Um−Tm]2− γ

∂2Um∂a2

Um−Tm+ (1 − γ)[∂Uf\∂a

Uf−Tf]2− (1 − γ)

∂2Uf

∂a2

Uf−Tf.

Since strong individual rationality holds (Um > Tm and Uf > Tf), a sufficient butnot necessary condition for −∂F

∂a > 0 is that Um and Uf are concave in a, which istrue in the Pareto area (see Appendix to Section 6.1) - the second order conditionholds.

Differentiate the first order condition with respect to γ and get that ∂F∂γ =

∂Um∂a

Um−Tm−

∂Uf∂a

Uf−Tf> 0. Therefore, using the second order condition, we have ∂a

∂γ =∂F∂γ

− ∂F∂a

> 0, and naturally the male gets less home duties the larger is his bargainingpower.

For the sharing effect the second-order effects are too complicated to yield ameaningful comparative static. However, in all our results the first order effect,i.e. that as sharing increases, the male wants to induce work effort from the female

37

and takes more home duties, dominates (see Figures 3 and 4 for an example) andfor reasonable perturbation of parameters we have ∂a

∂s > 0.What matters for the argument that gender based taxes change the implicit

bargaining power is that ∂a∂ tm

tf

< 0. However, the intuition may well be inspected

by changing one tax rate at the time.The redistribution, threat and cross redistribution effects follow from simple

inspection of the utilities under marriage and under autarky. That the threateffect dominates the redistribution effect can be established by differentiating Um

and Tm to obtain ∂Um∂tm

− ∂Tm∂tm

= −(s(1 − tm))am+2am−2 − (−(1 − tm)

φ+2φ−2 ) > 0 which

holds if (but not only if) s < 1 and am < φ. A weaker sufficient condition is thata single person takes less home duties than a married person, which we believeis a reasonable condition. This condition becomes sufficient and necessary for noresource sharing, s = 1.

That ∂Um∂tm

− ∂Tm∂tm

> 0 is ”almost” sufficient for ∂am∂tm

< 0 can be established as

follows. We want to show that ∂F∂tm

< 0. For this write ∂F∂tm

= γ−∂Um\∂a

[Um−Tm]2[∂Um

∂tm−

∂Tm∂tm

]+γ∂2Um∂a∂tm

Um−Tm+(1−γ)−∂Uf\∂a

[Uf−Tf ]2[∂Uf

∂tm]+ (1−γ)

∂2Uf∂a∂tmUf−Tf

. We have that ∂Um∂a > 0 and

∂Uf

∂a < 0. Now the term ∂Um∂tm

− ∂Tm∂tm

is positive because the threat effect dominates

the redistribution effect. The term ∂Uf

∂tmis the cross redistribution effect and it is

negative. The term ∂2Um∂a∂tm

by virtue of the Envelope and Young’s Theorems can be

written as ∂2Um∂a∂tm

= −s∂Em∂a and it is negative as earnings decrease with home duties.

Finally, the term ∂2Uf

∂a∂tm= −(1 − s)[∂Em

∂a − (1 − tm) ∂2E∂a∂tm

] may be either positiveor negative, depending on the elasticity of earnings with respect to home duties.If it is negative, which holds when s is not too high (say in Area II), then ∂F

∂tm< 0

as wanted. If it is positive, then ∂F∂tm

< 0 cannot be established analytically, butin our numerical results the last effect never dominates the three first effects. Thereason is that for s very large which is necessary for ∂Em

∂a − (1 − tm) ∂2E∂a∂tm

< 0to hold, (1 − s) times ∂Em

∂a − (1 − tm) ∂2E∂a∂tm

becomes negligible. In the absence ofsharing, s = 1, ∂F

∂tm< 0 always holds because the first and the second terms are

always negative, and therefore ∂a∂tm

< 0. ∂a∂tf

> 0 can be examined similarly.

Appendix to Section 6

We denote am = a and af = A− a. We calibrate the total number of home dutiesto be 20 (A = 10). This delivers elasticities of labor supply around .2 for the maleand .3 for the female under no resource sharing. The bargaining power γ is setat 3/4 because (i) with resource sharing and (ii) spouses being ”risk averse in a”,the allocation of resources is quite balanced. For example, with s = 1 (i.e. the

38

male willing to take as less home duties as possible) and z = .2 , the male extractsaround 60% of the marriage surplus. For s < 1 this is even smaller and changingthe bargaining power does not create significant variation in the results. Similarlyfor z (subject to maintaining strong individual rationality).

In drawing Figures 6-9 we keep constant public expenditure as a percentage ofGDP. The reason is that GDP falls quickly with a declining s (both spouses workless), and therefore holding constant the level of public expenditure G results inunmeaningful comparisons. G/GDP is set at 20%.

Even though 2am and 2af can take only integer values, for expository reasonswe discretize the total number of shocks A into a more continuous grid and treatthem as continuous variables when conducting comparative statics. Alternatively,we could increase A to create meaningful variations in am and af , but at theexpense of calibrating the elasticities and burdening the notation.

To solve the model we specify a grid for a and two grids on tm and tf . Afterderiving the Nash-bargaining solution as a function of the tax rates, we substitutea(tm, tf) in the social welfare function and search for the optimum set of taxesthat satisfy the government’s budget constraint.

Appendix to Section 6.1

We don’t have an analytic expression for the solution of the bargaining program.Working numerically and intuitively, the first point is that Um is not globallyconcave in a. Taking the second derivative with respect to a we have, for examplefor the male, that ∂2Um

∂a2 = La ln L[ 1a2 − lnL

a ] + La[− 2a3 + ln L[ 1

a2 + 1a(a−2)

]] + (1 −

s)(1 − tf )∂2Ef

∂a2 , where Ef = WfLf are the female’s earnings. While the first andthe second terms are negative, the third term is ambiguous and depends amongother things on the level of sharing. However, except for extreme levels of a (closeto 2 or close to A-2) and extreme levels of sharing (s close to 1/2), concavityis ensured. In particular, in the absence of sharing (s = 1), the first two termsyield an unambiguous concave indirect utility Um. In our numerical results, Um

is concave in a everywhere in the Pareto efficient area (i.e. when (20) holds, seeFigure 5). Similarly for the concavity of Uf . Since the Nash-bargained allocationsare by assumption Pareto efficient, concavity in the area of interest is assured, andthe bargaining solution is well defined.

Appendix to Section 6.2

The first order necessary condition for interior local optimum for the programdescribed in Section 6.2 is given by

39

∂Uf

∂tf+ ∂Um

∂tf+ ( ∂a

∂tf)[∂Um

∂a + ∂Uf

∂a ]∂Um∂tm

+ ∂Uf

∂tm+ ( ∂a

∂tm)[∂Um

∂a + ∂Uf

∂a ]=

Ef + tf∂Ef

∂tf+ tf

∂a∂tf

∂Ef

∂a

Em + tm∂Em∂tm

+ tm∂a∂tm

∂Em∂a

(27)

where Ej = WjLj are gross earnings. This condition says that at the optimumthe female over the male ratio of social marginal cost should equal the ratio ofmarginal revenues that the government can extract from each spouse respectively.Multiplying by 1−tm

1−tfboth sides we can rewrite the first order condition as

[ 11−tf

][∂Uf

∂tf+ ∂Um

∂tf] + [ 1

1−tf]( ∂a

∂tf)[∂Um

∂a + ∂Uf

∂a ]

[ 11−tm

][∂Um∂tm

+ ∂Uf

∂tm] + [ 1

1−tm]( ∂a

∂tm)[∂Um

∂a + ∂Uf

∂a ]=

[ 11−tf

][Ef + tf∂Ef

∂tf] + [ tf

1−tf] ∂a∂tf

∂Ef

∂a

[ 11−tm

][Em + tm∂Em∂tm

] + [ tm1−tm

] ∂a∂tm

∂Em∂a

(28)

While certainly not sufficient this condition can shed some light in the workingsof the solution. In the left hand side, the first terms in the numerator and thedenominator are the same as in the case of the exogenous bargaining problem (asin equation (25)). The second terms in the numerator and the denominator appearbecause the government desires to affect the allocation of home duties. The termin the brackets [∂Um

∂a + ∂Uf

∂a ] is common in the numerator and the denominator.This would have been the first order condition if the government could affect a

directly. Starting from a > A−a (i.e. the male getting less home duties) this termis negative because of decreasing returns of specialization (at least, in the Paretoarea). From the analysis in Section 5.3 and in this Appendix the term ∂a

∂tfin the

numerator is positive and the term ∂a∂tm

in the denominator is negative.Therefore, relative to the case with exogenous bargaining, the ratio of the

female’s to the male’s social marginal cost of taxation ∂Ω∂tf

\ ∂Ω∂tm

increases. Withendogenous bargaining and starting from γ > 1/2 it is relatively more costlyto tax the female than it is in the exogenous bargaining case. Every unit oftax revenues that the government raises from the female further deteriorates herimplicit bargaining power and results in a more gendered allocation. This intuitionlies in the heart of the tm > tf result in Section 6.2.

Things however are complicated by the fact that the ratio of marginal revenuesalso changes relative to the exogenous bargaining case. The difference stems fromthe last terms in the numerator and the denominator of the right hand side of (28).The term ∂a

∂tj

∂Ej

∂a measures the shift in the peak of the Laffer curve for spouse j dueto the shift in the intrahousehold allocation of resources. For example, increasingthe male’s tax rate results in less bargaining power for the male who has to ”settlein” with a smaller a. Then the male participates less in the labor market andbecomes less risk averse, per the intuition of Section 4. This increases his labor

40

supply elasticity, which poses an extra cost for the society since the governmentwants to tax the male. Relative to the exogenous bargaining case, the last termsin the numerator and the denominator in general raise the female over the maleratio of marginal revenues. The reason why this appears to be true is that foram > af we have that ∂Ef

∂a is greater than ∂Em∂a in absolute value because earnings

are concave in a. Also in our simulations ∂a∂tm

seems to be less responsive than∂a∂tf

due to the bargaining power of the male. If the ratio of the marginal revenuesincreases, then it is less easy to extract revenues from the male in the endogenousbargaining case. The simultaneous increase of the ratio of marginal costs andthe ratio of marginal revenues under endogenous bargaining, prohibits us fromcomparing the optimal solution tm

tfunder the two regimes.

Finally, the relationship between pre-gender based taxation inequality and thesharing parameter s can be examined by writing the inequality ratio as

I = (s(1 − t))am

am−2− A−am

A−am−2 (29)

The first point is that since for s = 1 and γ > 1/2 we always have am > A−am,we get that I(s = 1) > 1. Second, for a given level of t that raises revenues equalto G, let’s call K(s) = am(s)

am(s)−2 − A−am(s)A−am(s)−2 . Since a′m(s) > 0, we have that

K ′(s) < 0. The two opposite forces of s on I can been illustrated as follows. Forgiven K(s) < 0, a higher s decreases I because the female participates more inorder to balance the less sharing of resources that takes place in the family. Forgiven s(1 − t), a higher s causes K(s) to become more negative and this tendsto increase inequality I . This is because the male shares less resources with thefemale and “exerts” his bargaining power by choosing an even more unbalancedhome duties ratio. The two forces exactly cancel out at point sM = .92 in Figures6 and 7.

41

Table 1: Welfare effects of Gender Based Taxation with exogenous bargaining

Parameter values Endogenous ratios Gains (in %)

Focus Tax regime GGDP

am

af

σm

σfs Lm

Lf

τm

τf

Um

Uf

tmtf

Ω L τ GDP

G GBT 18% 1.83 0.50 0.95 1.05 0.98 1.16 1.35 0.49 0.50 0.71 1.07single 1.07 1.07 1.32 1GBT 22% 1.83 0.50 0.95 1.06 0.98 1.16 1.32 0.87 0.63 0.99 1.41single 1.09 1.09 1.36 1

σm

σfGBT 20% 1.83 0.50 0.95 1.05 0.98 1.16 1.34 0.65 0.56 0.84 1.23

single 1.08 1.08 1.34 1GBT 20% 2.58 0.33 0.95 1.09 0.97 1.31 1.66 2.42 1.96 2.96 4.30single 1.16 1.16 1.72 1

Notes: In the first three rows elasticities are σm = 0.1 for the male and σf = 0.2 for the female. For the last row we haveσm = 0.089 and σf = 0.267 respectively. For later reference, we note that in all cases we keep the total number of ”shocks”A = af + am constant. For the version of the model with CRRA subutility for consumption, for elasticities σm = 0.07 andσf = 0.28 we find the ratio of optimal taxes to be tm

tf= 1.62, welfare gains of approximately 0.26%, employment gains of

around 0.93%, gains in training of 0.42%, and GDP gains of 1.24%.

42

Figure 1: The Labor Market

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Labor

Wag

e R

ate

Males

Females

43

Figure 2: The Effects of Taxes on the Labor Market

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Labor

Wag

e R

ate

low tax

high tax

Low Tax

High Tax

44

Figure 3: Sharing of Resources and Allocation of Shocks

4 4.5 5 5.5 6 6.5 7 7.50.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

a (m)

Util

ity (

mal

e)

s=1

s=0.9

s=0.8

45

Figure 4: The Bargaining Solution

0.65 0.7 0.75 0.8 0.85 0.9 0.950.45

0.5

0.55

0.6

s

Fra

ctio

n o

f Hom

e D

utie

s to

Fem

ale

tm/tf = 0.85

tm/tf = 1

tm/tf=1.15

46

Figure 5: Ungendered Equilibrium is the First Best

3 3.5 4 4.5 5 5.5 6 6.5 70.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

a

Util

ity

U (m)U (f)

Pareto Efficient Allocations

47

Figure 6: tmtf

, σm

σf- s; γ = 3/4, z = 0.2, G

GDP= 20%

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.5

1

1.5

s

Tax

(m

) / T

ax (

f)

Ela

stic

ity (

m)

/ Ela

stic

ity (

f)

Tax (m) / Tax (f)

El (m) / El (f) : GBT

El (m) / El (f) : Single

I II III

48

Figure 7: Wage Ratios - s; γ = 3/4, z = 0.2, GGDP

= 20%

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.8

0.85

0.9

0.95

1

1.05

1.1

s

Wag

e (m

) / W

age

(f)

W (m) / W (f) : GBT

W (m) / W (f) : Single

I II III

49

Figure 8: Gains - s; γ = 3/4, z = 0.2, GGDP

= 20%

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

s

Gai

n %

Welfare Gain %

GDP Gain %

Employment %

III II I

50

Figure 9: Both Spouses May Be Better Off with GBT

0.7 0.72 0.74 0.76 0.78 0.8 0.82−2

−1

0

1

2

3

4

5

s

Gai

n U

tility

%

Gain % U (m)

Gain % U (f)

51

Figure 10: Social Welfare Indifference Curves and Revenue Constraint

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

tf

t m

Revenue Function (=G)

Welfare Indifference Curve

45 degree line

52